How could I transform the vertices of my mesh to appear in Perspective view, as it does in Orthographic view?
This sounds like a job for math, python, and bmesh.  You'll have to have an understanding of the mathematics behind orthographic and perspective projections so that you can compute replacement coordinates for the vertices.

Once you have the formula, you can write python which iterates through the vertices of the mesh and rewrites their coordinates.  Something like this:

obj = bpy.context.active_object
for v in obj.data.vertices:
    v.co = funkify(v.co)